Why does an electric light bulb nearly always burn out just as you turn on the light, almost never while the light is shining?

An electric light bulb is not an ideal resistor.  This
means that as the filament in the light bulb heats up, its resistance changes.  When it
is cool, the resistance is the lowest.  This means that when the switch is flipped to
turn it on, the current is larger initially and then decreases as the filament heats up
and the resistance increases.  The sudden change in temperature due to the current can
cause the brittle filament to break.  Once the bulb is shining, the temperature is
staying relatively constant and the filament is unlikely to
break.